A geometric approach to the transfer problem for a finite number of traders
aa r X i v : . [ q -f i n . E C ] J a n A geometric approach to the transfer problem for a finitenumber of traders
Tomohiro UchiyamaNational Center for Theoretical Sciences, Mathematics DivisionNo. 1, Sec. 4, Roosevelt Rd., National Taiwan University, Taipei, Taiwan email:[email protected]
Abstract
We present a complete characterization of the classical transfer problem for an exchangeeconomy with an arbitrary finite number of traders. Our method is geometric, using anequilibrium manifold developed by Debreu, Mas-Colell, and Balasko. We show that for aregular equilibrium the transfer problem arises if and only if the index at the equilibriumis −
1. This implies that the transfer problem does not happen if the equilibrium is Walrastatonnement stable. Our result generalizes Balasko’s analogous result for an exchangeeconomy with two traders.
Keywords: international trade, transfer problem, general equilibrium, equilibrium manifold,index theorem, economic stabilityJEL classification: D51, F20
In this paper, we study the following classical transfer problem (Samuelson, 1952, 1954):
Problem 1.1.
Suppose that in the world there are n countries trading l goods where n and l are arbitrary natural numbers. Then does it happen that after a country, say country A, givesaway some of her endowment to other countries, country A’s utility level goes up? If it happens,characterize it. Samuelson considered an exchange economy with n = 2 and l = 2 (with or without varioustrade impediments). He also hinted that (without proof) the transfer problem is closely relatedto the economic stability. Balasko verified that Samuelson’s intuition was correct. He showedthat in a smooth exchange economy (without any trade impediments) the tranfer problem arisesat a regular equilibrium if and only if the index at the equilibrium is −
1, in other words, thispathology happens only when the equilibrium is locally Walras tatonnement unstable. Usingthe theory of equilibrium manifold, Balasko initially showed this for n = 2 and l = 2 (Balasko,1978), then for n = 2 and an arbitrary finite number l (Balasko, 2014).The main purpose of this paper is to extend Balasko’s result for arbitrary finite numbers n and l . Roughly speaking, we show that Theorem 1.2.
For a smooth exchange economy with arbitrary finite numbers of traders (coun-tries) and goods (without any trade impediments), the transfer problem arises at a regularequilibrium if and only if the index at the equilibrium is − . In particular, the transfer problemarises only when the regular equilibrium is locally Walras tatonnement unstable. We denote by N , R , and R > the set of natural numbers, the set of real numbers, and the setof positive real numbers respectively. Fix l ∈ N . We set the commodity space X := R l> (thestrictly positive orthant of R l ). We set the l -th goods as a a numeraire: if p = ( p , · · · , p l ) ∈ X is a price vector, then p l = 1. We assume that all prices are strictly positive. The set of pricevectors is S := R l − > × { } . We have n ∈ N traders (countries). Each trader i ∈ { , · · · , n } isendowed with a goods vector ω i ∈ X . We assume that our economy is an exchange economywith no production: the total resources r = P ni =1 ω i of the economy is fixed. We write Ω = { ω = ( ω , · · · , ω n ) ∈ X n | P ni =1 ω i = r } for the space of endowment of the economy.We assume that each trader’s preferences is represented by a smooth utility function u i : X → R . For z i ∈ X , we write I z i := { y i ∈ X | u i ( y i ) = u i ( z i ) } (the indifference surface fortrader i going through the point z i ). We assume that each u i satisfies the following standardhypotheses for the differential equilibrium analysis (Mas-Colell, 1985), (Balasko, 2009). Forany x i ∈ X , (i) smooth monotonicity: Du i ( x i ) ∈ R l> ; (ii) smooth strict quasi-concavity:the Hessian D u i ( x i ) is negative definite on the tangent plane to I x i ; (iii) I x i is closed in X .Moreover, we extend u i to x i = 0 by setting u i (0) = inf x i ∈ X u i ( x i ).Given a price vector p ∈ S and an budget w i = p · ω i ∈ R > , each trader maximizes theutility u i ( x i ) under the budget constraint p · x i ≤ w i . Then there exists a unique solution tothis utility maximization problem, and we obtain a smooth demand function f i : S × R > → X satisfying Walras’s law: p · f i ( p, w i ) = w i . Note: Do not confuse w i with ω i . We are followingDebreu’s (and Balasko’s) notation (Debreu, 1970), (Balasko, 2014).2 The equilibrium manifold
For p ∈ S and ω ∈ Ω, let z ( p, ω ) := P ni =1 f i ( p, p · ω i ) − r . Then z ( p, ω ) is the (social) excessdemand associated with ( p, ω ). It is clear that the function z : S × X → R l is smooth. Definition 3.1.
We call ( p, ω ) ∈ S × X is an equilibrium if z ( p, ω ) = 0. The subset E of S × Ωdefined by z ( p, ω ) = 0 is called the equilibrium manifold .Note that the equilibrium manifold E is a real smooth manifold in the usual sense in differ-ential geometry (Milnor, 1965), (Guillemin and Pollack, 1974). See (Balasko, 2009, Prop. 2.4.1)for a proof.For ( p, ω ) ∈ S × X , we denote by z − l ( p, ω ) ∈ R l − the vector defined by the first l − z ( p, ω ). Since each trader satisfies Walras’ law at an equilibrium, we have z ( p, ω ) = 0 if and only if z − l ( p, ω ) = 0. (Thus it is enough to consider z − l ( p, ω ).) Let p ∈ S .We write p − l for the vector defined by the first l -coordinates of p . (The l -th coordinate is fixedby the numeraire assumption anyway.) Recall that an equilibrium ( p, ω ) ∈ E is regular if theJacobian matrix J ( p, ω ) := Dz − l ( p, ω ) /Dp − l is invertible (Mas-Colell, 1985) (Balasko, 2009).Suppose that ( p, ω ) ∈ E is a regular equilibrium. Then by applying the implicit functiontheorem to the equation z − l ( p, ω ) = 0, we can express p in terms of ω . Then by (Balasko,2011, Prop. 7.2), there exist a neighborhood U of ω , a neighborhood V of ( p, ω ), and a smoothmap s : U → S such that the map σ : U → V defined by σ ( x ) = ( s ( x ) , x ) is a diffeomorphismbetween U and V . Following (Balasko, 2014), we call σ (or s ) the local equilibrium selectionmap (the local equilibrium price selection map) associated with ( p, ω ). Now we are ready tostate the transfer problem in a precise way: Definition 3.2.
We say that there is a transfer problem at a regular equilibrium ( p, ω ) ∈ E ifthere exists an endowment vector ω ′ ∈ U and a trader i ∈ { , · · · , n } such that1. ω i ≤ ω ′ i and ω i = ω ′ i ,2. u i ( f i ( s ( ω ′ ) , s ( ω ′ ) · ω ′ i )) > u i ( f i ( s ( ω ) , s ( ω ) · ω i ))where s : U → S is the local equilibrium price selection map associated with ( p, ω ) definedabove.Note that the definition of the transfer problem at ( p, ω ) ∈ E requires ( p, ω ) to be regular(since otherwise we do not have the function s ). However this is not a strong restriction: theset of regular equilibria in E is open with the full measure of E ; see (Balasko, 2011, Prop. 8.10). In this short section, we review the definition of the index at an equilibrium ( p, ω ) andits relationship with economic stability to make this paper self-contained. Our references hereare (Arrow and Hahn, 1971), (Hirsch et al., 2004), and (Guillemin and Pollack, 1974). Recallthat the index of a regular equilibrium ( p, ω ) ∈ E is 1 (or −
1) if the sign of ( − l − det( J ( p, ω ))is positive (or negative respectively). It is well-known that if ( p, ω ) ∈ E is a Walras tatonnementlocally stable equilibrium, then all eigenvalues of J ( p, ω ) must have strictly negative real parts.Thus the index of such ( p, ω ) ∈ E must be 1. Note that the converse fails: there exists anequilibrium ( p, ω ) ∈ E such that the index of ( p, ω ) is 1 but ( p, ω ) is not Walras tatonnementlocally stable. See the references above for more on this.3
A geometric approach to equilibrium analysis
In this section, we follow the argument in (Balasko, 2014, Sec. 3) closely. Define the price-income space H ( r ) associated with the fixed social endowment r by H ( r ) := { ( p, w ) ∈ S × R n> | P ni =1 w i = p · r } where w is the income vector ( w , · · · , w n ). From the definition, it is clear that H ( r ) is an ( l + n − S × R n> . We set the coordinates for H ( r ) by( p , · · · , p l − , w , · · · , w n − ) = ( p − l , w − n ) ∈ R l + n − > . Note that we omitted the last coordinateof w since w n is determined by w n = p · r − P n − i =1 w i .Let B ( r ) := { ( p, w ) ∈ H ( r ) | P ni =1 f i ( p, w i ) = r } . We call B ( r ) the section manifold .This is a smooth ( n − H ( r ) by (Balasko, 2009, Prop. 5.4.1). Inthe following, we give a concrete parametrization of B ( r ) that comes handy for our geometricequilibrium analysis.Let P ( r ) be the set of Pareto optimal allocations. First, we parametrize P ( r ). Let U ( r ) bethe set of vectors ( u , · · · , u n − ) = u − n ∈ R n − of the first n − U ( r ) := { u − n ∈ R n − | there exists ( x , · · · , x n − ) ∈ X n − such that u = u ( x ) , · · · , u n − = u n − ( x n − ) and n − X i =1 x i ≤ r } . Given u − n ∈ U ( r ) we obtain a Pareto optimal allocation x = ( x , · · · , x n ) ∈ X n by solving thefollowing constrained optimization problem:max x ∈ X n u n ( x n ) subject to u − n = u − n ( x − n ) for some x − n ∈ U ( r ) and n X i =1 x i = r. We write x ( u − n ) = ( x ( u − n ) , · · · , x n ( u − n )) for the solution of this optimization problem.Under our assumptions on smooth preferences, there exists a unique price vector p ( u − n ) ∈ S that supports x ( u − n ). Note that p ( u − n ) is parallel to the gradiant vectors Du i ( x i ( u − n )). Wecan express all Pareto optimal allocations as x ( u − n ) by varying u − n in U ( r ).Here is a crucial observation: the point M ( u − n ) := ( p ( u − n ) , p ( u − n ) · x ( u − n ) , · · · , p ( u − n ) · x n ( u − n )) ∈ H ( r ) belongs to B ( r ) for any u − n ∈ U ( r ). Conversely, any point ( p, w ) ∈ B ( r ) isassociated with a unique point u − n ∈ U ( r ) via u − n = ( u ( f ( p, w ) , · · · , u n − ( f n − ( p, w n − ))).Thus, u − n ∈ U ( r ) parametrizes the section manifold B ( r ) as well as the set of Pareto optimalallocations P ( r ).For the map M : U ( r ) → B ( r ), we write ∂M∂u i for the partial derivative of M with respectto u i for i ∈ { , · · · , n − } . We set a positive orientation for B ( r ) via ( ∂M∂u , · · · , ∂M∂u n − ). For u − n ∈ U ( r ), the vector ∂M∂u i ( u − n ) represents the direction of increasing the utility level fortrader i in a neighborhood of M ( u − n ).For ω ∈ Ω, we define the budget space A ( ω ) associated with the endowment ω by A ( ω ) := { ( p, w ) ∈ H ( r ) | w i = p · ω i for i ∈ { , · · · , n − }} . Then A ( ω ) is a linear space being theintersection of hyperplanes defined by w i = p · ω i in the hyperplane H ( r ).It is easy to see that ( p, ω ) ∈ S × Ω is an equilibrium if and only if ( p, p · ω , · · · , p · ω n ) ∈ A ( ω ) ∩ B ( r ). The point is that we split the equilibrium analysis into the analysis of twoseparate parts A ( ω ) and B ( r ). The first part A ( ω ) is a linear space. Thus all nonlinearities ofthe equilibrium equation z ( p, ω ) = 0 is captured by B ( r ). Note that B ( r ) does not depend on ω : all ω -terms are in A ( ω ).Using the coordinate system ( p − l , w − n ) for H ( r ) and the equations w i = p · ω i , we see that A ( ω ) is parallel to the ( l − { a ( ω ) , · · · , a l − ( ω ) } (in4erms of this coordinate system) is given by a ( ω ) := (1 , , · · · , , ω , ω , · · · , ω n − ) ,a ( ω ) := (0 , , , · · · , , ω , ω , · · · , ω n − ) , · · · a l − ( ω ) := (0 , · · · , , , ω l − , ω l − , · · · , ω l − n − ) , where ω ji is trader i ’s endowment of goods j. We set the positive orientation of A ( ω ) via ( a ( ω ) , · · · , a l − ( ω )). Clearly the set of vectors a i ( ω ) is linearly independent, so the dimension of A ( ω ) is l − p, ω ) ∈ E . Then ( p, p · ω , · · · , p · ω n ) is the correspondingpoint in the price-income space H ( r ). Let u i ( p, ω ) := u i ( f i ( p, p · ω i )) for i ∈ { , · · · , n − } .Since ( p, ω ) is regular, smooth submanifolds A ( ω ) and B ( r ) of H ( r ) intersect transversally at( p, p · ω , · · · , p · ω n ). By the celebrated transversality theorem (Mas-Colell, 1985, Chap.1.I),this is equivalent to∆( p, ω ) := det (cid:18) a ( ω ) , · · · , a l − ( ω ) , ∂M∂u ( u − n ( p, ω )) , · · · , ∂M∂u n − ( u − n ( p, ω )) (cid:19) = 0 . Here is the main result of this section.
Proposition 5.1.
Let ( p, ω ) ∈ E be regular. Then the index of ( p, ω ) is (or − ) if ∆( p, ω ) is positive (or negative respectively).Proof. Although we could prove the proposition by a (not illuminating) tedious direct com-putation, we avoid it by utilizing the parametrization of B ( r ) via U ( r ) above. The followingargument shows that ∆( p, ω ) has the opposite sign of J ( p, ω ) for any regular ( p, ω ) ∈ E , whichis sufficient for our purpose.For u − n ∈ U ( r ). let M ( u − n ) = ( p ( u − n ) , w ( u − n ) , · · · , w n ( u − n )) ∈ B ( r ). Now let ω i ( u − n ) := f i ( p ( u − n ) , w i ( u − n )) for i ∈ { , · · · , n } , and write ω ( u − n ) for ( ω ( u − n ) , · · · , ω n ( u − n )). Then( p ( u − n ) , ω ( u − n )) is a no-trade equilibrium. By (Balasko, 2011, Prop. 8.2) every no-trade equi-librium is regular, so ( p ( u − n ) , ω ( u − n )) is regular.Note that A ( ω ( u − n )) varies continuously as u − n moves around in U ( r ). Thus the function δ : U ( r ) → R defined by δ ( u − n ) := ∆( p ( u − n ) , ω ( u − n )) is continuous. We see that δ never takethe value 0 for any u − n ∈ U ( r ) since every no-trade equilibrium is regular. So it is enough tocheck the sign of δ for any particular value of u − n since U ( r ) is connected. (We are using thesame argument as in the proof of (Balasko, 2014, Lem. 1).)Let ω = (0 , , · · · , , r ) ∈ X n . Then u − n ( ω ) := ( u (0) , u (0) , · · · , u n − (0)). Thus∆ ( p ( u − n ( ω )) , ω ) :=det (cid:18) e , e , · · · , e l − , ∂M∂u ( u − n ( ω )) , · · · , ∂M∂u n − ( u − n ( ω )) (cid:19) where e i is the column vector whose i -th coordinate is 1 and all other coordinates are zero.Let ∆ ′ ( p ( u − n ( ω )) , ω ) be the submatix of ∆ ( p ( u − n ( ω )) , ω ) formed by the last ( n −
1) rowsand columns of ∆ ( p ( u − n ( ω )) , ω ). Then δ ( u − n ( ω )) > ′ ( p ( u − n ( ω )) , ω )) > . Note that the vector ∂M∂u i ( u − n ( ω )) for i ∈ { , · · · , n − } is the direction of increasing trader i ’s utility level holding other trader’s utility level fixed at u j (0) for all j ∈ { , · · · , n − }\{ i } .5ince the income of all traders i ∈ { , · · · , n − } is 0 at ( p ( u − n ( ω )) , ω ), we must have ∂M∂u i ( u − n ( ω )) = (0 , , · · · , , x, , · · · ,
0) where x is the i -th component and x ∈ R > . Therefore we clearly have det (∆ ′ ( p ( u − n ( ω )) , ω )) >
0, and we are done.
Remark . We have shown that the index at ( p, ω ) ∈ E is same as the intersection number of A ( ω ) and B ( r ) at the corresponding point ( p, p · ω , · · · , p · ω n ) ∈ H ( r ); see (Guillemin and Pollack,1974, p.96) for the definition of the intersection number of two smooth manifolds. Let i ∈ { , · · · , n − } . Let ω, ω ′ ∈ Ω. Following (Balasko, 2014, Sec. 4), we say that thehyperplane A ( ω ) lies below the hyperplane A ( ω ′ ) for trader i if p · ω i < p · ω ′ i is satisfied for all p ∈ S . It is clear that A ( ω ) lies below A ( ω ′ ) if and only if ω i (cid:8) ω ′ i ; see (Balasko, 2014, Lem. 2)for a proof.Now, we interpret the local equilibrium price selection map in Definition 3.2 using A ( ω ) and B ( r ). If ( p, ω ) ∈ E is regular, then A ( ω ) and B ( r ) intersect transversally at ( p, p · ω , · , p · ω n )in the price-income space H ( r ). Then if ω ′ ∈ Ω is sufficiently closed to ω , A ( ω ′ ) ∩ B ( r ) containsa unique point in some small neighborhood of ( p, p · ω , · · · , p · ω n ) ∈ H ( r ). Thus, in someneighborhood of ω we obtain a map s : ω ′ → ( p ( ω ′ ) , p · ω ′ , · · · , p · ω ′ n ) such that s ( ω ) = ( p, p · ω , · · · , p · ω n ). Now composing this map with the projection ( p ( ω ′ ) , p · ω ′ , · · · , p · ω ′ n ) → p ( ω ′ )gives the local equilibrium price selection map in Definition 3.2. Theorem 6.1.
A regular equilibrium ( p, ω ) features the transfer problem if and only if theindex of ( p, ω ) is − . In particular, if ( p, ω ) is Walrus tatonnement locally stable, the transferproblem does not arise.Proof. Let ( p, ω ) ∈ E with b ( p, ω ) := ( p, p · ω , · · · , p · ω n ) ∈ A ( ω ) ∩ B ( r ). Let u i ( p, ω ) = u i ( p, p · ω i ) for i ∈ { , · · · , n − } . Then M ( u ( p, ω ) , · · · , u n − ( p, ω )) = ( p, p · ω , · · · , p · ω n ) ∈ H ( r ) ( M is defined in Section 5). Let ω = (0 , , · · · , , r ) ∈ Ω and ω = ( r, , , · · · , ∈ Ω. Set t ( ω ) := M ( u (0) , · · · , u n − (0)) ,t ( ω ) := M ( u ( r ) , · · · , u n − (0)) . Since B ( r ) is a smooth connected manifold and parametrized by u − n ∈ U ( r ), there exist curves C (and C ) from t ( ω ) to b ( p, ω ) (and from b ( p, ω ) to t ( ω ) respectively) where points on C (those on C ) correspond to the utility level of trader 1 lower than or equal to u ( p, ω ) (higherthan or equal to u ( p, ω ) respectively).If the intersection number of A ( ω ) and B ( r ) is − b ( p, ω ), there exists a neighborhood U ⊂ H ( r ) of b ( p, ω ) such that all points in U ∩ C is above A ( ω ) for trader 1. Likewise, allpoints in U ∩ C is below A ( ω ).Let U ′ be an open neighborhood of ω where the local equilibrium price selection map s : U ′ → P is defined. Without loss, we choose U ′ such that the image of the projection of U onto Ω is contained in U . Then if ω ′ ∈ U ′ and ω ′ (cid:8) ω , A ( ω ′ ) lies below A ( ω ) for trader 1. So b ( s ( ω ′ ) , ω ′ ) lies below A ( ω ) for trader 1. Then b ( s ( ω ′ ) , ω ′ ) belongs to the curve C . Thus wehave u ( f ( s ( ω ′ ) , s ( ω ′ ) · ω ′ )) > u ( f ( s ( ω ) , s ( ω ) · ω )).Similarly we can show that if the intersection number of A ( ω ) and B ( r ) is 1 at b ( p, ω ), then u ( f ( s ( ω ′ ) , s ( ω ′ ) · ω ′ )) < u ( f ( s ( ω ) , s ( ω ) · ω )) for ω ′ ∈ U ′ with ω ′ < ω .6ote: although the proof shows there exists the transfer problem which improve trader 1’sutility level after he/she gives away some of his/her endowment, there is nothing special fortrader 1. A similar argument works for any trader i . Remark . Suppose that we are at a no-trade equilibrium ( p, ω ) ∈ E . In (Balasko, 2011),Balasko showed that the set of regular equilibria is partitioned into a finite number of path-connected components and the index is constant in each component. Moreover he showed thatthe set of no-trade equilibria is contained in the unique component of index 1. Therefore thetransfer problem arises only for a large amount of trades. Open Problem 7.1.
Our model does not have a production sector. It would be interesting tosee how the result changes if we introduce a production sector into the model.
Open Problem 7.2.
We have avoid the issue of trade costs. It is natural to ask: what happensif we introduce various kinds of trade impediments? Some discussions are in (Samuelson, 1954).
Acknowledgements
This research was supported by a postdoctoral fellowship at the National Center for Theo-retical Sciences at the National Taiwan University.
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